Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 73

Lemma4.2

M
∑

M

( )

i

a = M , ∀ ∈ i {1, 2,  , M }.

.

mn

.

m = 1 n = 1

Lemma4.3

M
∑

( )

i

a mn = 1, ∀ m n , ∈ {1, 2,  , M }.

. .

i = 1

Lemma4.4

∑ ∑

N
L L

( ) i ( ) i 0

a − a = ± ≡ C

mn mn row

− − 1

g

ξ ∈ Z ξ ∈ Z ′

.

n n

where. Z is. Z odd .or. Z even ,.and. N 0 .is.the.number.of.possible.pasting.positions.in.
ξ .such.that.the.pasted.region.contains.don't-care.bits.(*).only.

Lemma4.5

N
L L

∑ ∑

( ) i ( ) i all

a − a = ± ≡ C

mn mn col

2

( − − 1)

g

.

ξ ∈ Z ξ ∈ ′ Z

m m

where. Z is. Z odd .or. Z even ,.and. N all .is.the.number.of.possible.pasting.positions.
in.

ξ .such.that.the.pasted.region.includes.all.the.actual.bits.

Lemma4.6

− 1 < C + C < 1.

row col

Lemma4.7

i i

Consider.a.distribution.matrix. A

( )

= a

( )

.with.size. M ×. M ;.one.has

{ } mn

∑ ∑

M

( ) i

Q = a =

4 (1

+ C + C );

1 mn row col

.

ξ ∈ Z ξ ∈ Z

m n

∑ ∑

M

( ) i

Q = a =

4 (1

− C − C );

2 mn row col

.

ξ ∈ Z ′ ξ ∈ Z ′

m n